Multisensor triplet Markov fields and theory of evidence

64
q y ðxÞ Z Y s2S
" !#
ðpðy s jx s ÞÞ= X u2U
pðy s jx s Z uÞ
(we will write q y ðxÞf Q s2S pðy s jx s Þ in the following), we have
p(xjy)Z(p4q y )(x) (where p is the Markov distribution of X).
This provides numerous possibilities of extension of p(xjy), by
replacing in p4q y either p or q y by a mass function. In this
section we shall show that when the Markov probability
distribution p is replaced in p4q y by a ‘Markov’ mass function
M, the M4q y can be seen as a marginal distribution of a
particular TMF. An important consequence of this observation
is that M4q y can be used to estimate X from Y. Now, let us
consider a set of classes U and the power set P(U). Let nZ
Card(S) and M 0 be a bba defined on [P(U)] n by
" #
M 0 ðAÞ Z g exp K X c2C
j c ðA c Þ
(3.3)
where C is the set of cliques corresponding to a given
neighborhood defining the Markovianity, AZ(A s ) s2S , and
A c Z(A s ) s2c . Such a bba will be called ‘evidential Markov
field’ (EMF). We see that an EMF extends the classical Markov
field, the latter being obtained when M 0 is null outside {{u 1 },
.,{u k }} n . Both EMF M 0 and the probability distribution q y
(given by pðyjxÞZ Q s2S pðy s jx s Þ) define a ‘hidden’ EMF
(HEMF). We have the following result.
Proposition 3.1. Let M 0 be an EMF defined on [P(U)] n by
(3.3), and M 1 a probability over U n defined from the observed
field YZy2R n by M 1 ðxÞZq y ðxÞf Q s2S pðy s jx s Þ. Let LZP(U)
and D3U!L such that (u,l)2D if and only if u2l.
Then the probability distribution MZM 0 4M 1 is the
marginal distribution pðxjyÞZ P u2½PðUÞŠn pðx; ujyÞ, where p(x,
ujy) is a Markov distribution obtained from the TMF TZ(X,U,
Y), the distribution of which is defined on (D!R) n by (3.1),
with
4 c ðt c Þ Z 4 c ðx c ; u c ; y c Þ
(
Z j cðu c Þ; for Card ðcÞO1;
j c ðu c ÞKLogðpðy s jx s ÞÞ; for c Z fsg
Proof. We have
MðxÞ Z ðM 0 4M 1 ÞðxÞf X "
exp K X # Y
j c ðu c Þ pðy s jx s Þ
x2u c2C s2S
" #
Z X exp K X j c ðu c Þ C X logðpðy s jx s ÞÞ
x2u c2C
s2S
(3.4)
In the sum above xZ(x s ) 2S is fixed and uZ(u s ) 2S varies in
[P(U)] n in such a way that x s 2u s for each s2S, which means
that uZ(u s ) 2S varies in such a way that (x,u)2D n . This means
W. Pieczynski, D. Benboudjema / Image and Vision Computing 24 (2006) 61–69
that M(x) can be written as
"
X
MðxÞf
exp K X j c ðu c Þ C X #
logðpðy s jx s ÞÞ
u=ðx;u;yÞ2D n !R n c2C s2S
Z
X
exp½4 c ðx c ; u c ; y c ÞŠ
u=ðx;u;yÞ2D n !R n
which completes the proof.
,
Example 3.1.. One possible application of the HEMF above is
inspired by the successful use of the similar hidden evidential
Markov chains in the situations where the unknown process has
unknown parameters and is not stationary [12]. Thus, let us
consider the problem of segmenting an observed image YZy
into two classes UZ{u 1 ,u 2 }, and let us assume that the hidden
class image XZx is strongly heterogeneous and can hardly be
modeled by a stationary Markov field. As HEMF is a particular
TMF, the parameter estimation method proposed in [1] can be
applied and thus unsupervised segmentation is workable. One
can then compare two unsupervised methods: the classical
HMF based method, and the new HEMF one.
Two examples of unsupervised segmentation performed
with the classical HMF and the proposed HEMF are presented
in Fig. 1. We consider two classes UZ{u 1 ,u 2 }, and assume
that both HMF and HEMF are Markovian with respect to the
four nearest neighbors. Therefore, we have an EMF on (L) n ,
with LZP(U)Z{{u 1 },{u 2 },{u 1 ,u 2 }}Z{l 1 ,l 2 ,l 3 }, the distribution
of which is given by (3.3). We then consider a simple
energy function given by the following potential functions. For
horizontal cliques cZ(t,s), we take j c (l 1 ,l 2 )ZKa 1H , j c (l 1 ,-
l 3 )ZKa 2H , j c (l 2 ,l 3 )ZKa 3H , and j c (l 1 ,l 1 )Zj c (l 2 ,l 2 )Z
j c (l 3 ,l 3 )Z0. The same is done for vertical cliques, with a IV ,
a 2V a 3V instead of a 1H , a 2H , and a 3H . Moreover, j c is null for
the cliques singletons.
According to Proposition 3.1, we have to consider the triplet
Markov field defined by
(
4 c ðt c Þ Z 4 c ðx c ; u c ; y c Þ Z j cðu c Þ; for c Z ðs; tÞ;
Klogðpðy s jx s ÞÞ; for c Z fsg
where (x s ,u s )areinDZ{(u 1 ,{u 1 }),(u 1 ,{u 1 ,u 2 }),(u 2 ,{u 2 }),(u 2 ,{-
u 1 ,u 2 })}Z{d 1 ,d 2 ,d 3 ,d 4 }. Thus, putting v s Z(x s ,u s ), we have a
standard hidden Markov field (V,Y) and for each s2S, the
probability p(v s jy) onDZ{d 1 ,d 2 ,d 3 ,d 4 } can be estimated by using
the Gibbs sampler. The estimates obtained this way enable us to
compute p(x s Zu 1 jy)Zp(v s Zd 1 jy)Cp(v s Zd 2 jy) and p(x s Z
u 2 jy)Zp(v s Zd 3 jy)Cp(v s Zd 4 jy), which are then used to perform
the Bayesian MPM segmentation. Concerning the classical hidden
Markov field used, the potential functions are j c (u 1 ,u 2 )ZKa H for
horizontal cliques, j c (u 1 ,u 2 )ZKa V for vertical cliques,
j c (u 1 ,u 1 )Zj c (u 2 ,u 2 )Z0 for horizontal and vertical cliques, and
j c is null for the cliques singletons. The estimates of all parameters
in both models are presented in Table 1, and the different images
are presented in Fig. 1.